Capstone 6940
Cox regression is a popular regression modeling method used to predict survival rates given certain covariates.
It assumes that the effects of different variables on the outcome, survival, are constant over time.
“Survival” can refer to the development of a symptom, time to relapse after remission, or as a time to death [1].
Cox regression model is based on the hazard function h_x(t) with covariates at time t given by:
h_x(t)=h_0(t)\exp(\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_n) h_x(t) = hazard \ function\ h_0(t) = Baseline \ hazard \ function\
\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_n = linear\ combination\ of \ covariates\ and\ their\ coefficients
The hazard function is the probability that an individual will experience an event (death) within a certain time interval [1].
If the risk factor is binary, the result can be interpreted as the estimated increase in HR in patients with the risk factor vs. those without [2].
If the risk factor is continuous, the HR is interpreted as an increase/decrease in the hazard rate of death due to a 1 unit increase in the variable [2].
The assumption of a constant relationship between dependent and explanatory variables is called proportional hazards.
Graphical strategies to assess proportionality assumption
Kaplan-Meier Survival Distribution S(t): Plot S(t) as a function of survival time for each level covariate.
plot the function log(-log(S(t))) as a function of the log survival time (log represent natural log).
Schoenfeld Residuals
Failing to meet the assumption of proportional hazards means that the effects between dependent and explanatory variables are not constant over time.
Time-varying covariates (coefficients) are used when a covariate changes over time during the follow-up period [3].
Internal time-varying coefficients are affected by survival status and include values that are generated by the subject [3].
External time-varying coefficients are pre-determined and external to the subject under study [3].